Reading this first essay by S. L. Glashow was a moment of pure pleasure. As witty as ever, it was an excellent stimulator of thought and imagination. There is hardly anything I would have liked to add and certainly nothing I would like to subtract. Since I do not have Glashow’s erudition, I will limit myself to a couple of examples from the Greek period he evokes for which I may have a more direct access to some sources.

I start with Democritus of Abdera (ca. 460–370 BCE), an almost legendary figure in ancient Greece. According to ancient sources his œuvre was immense, but only small fragments of it are known to us. Together with his master Leucippus he is considered to be the father of atomic theory. For Democritus the basic constituents of matter are the atoms and the vacuum—i.e., the empty space between the atoms. I copy from one fragment: “Νόμω γάρ χροιή, νόμω γλυκύ, νόμω πικρόν, ετεή δ ́ άτομα καί κενόν,” which in free translation says that “Laws determine the tone, the sweetness or the bitterness, but everything consists of atoms and empty space.” I have no idea whether Democritus meant anything in particular with the term κενόν—vacuum or empty space—and Glashow may want to comment on it in one of his next essays. The term vacuum has had many meanings in the history of science including our present ideas on the structure of the world. I would be curious to learn its evolution through the centuries.

My second remark refers to Euclid from Alexandria (ca. 300 BCE). He is mostly known for his Elements, the first axiomatic formulation of geometry and, probably, the first axiomatic system ever conceived. It was a real breakthrough in science in general and mathematics in particular. Results are obtained rigorously from a set of axioms following strict logical rules. It has influenced all western science, sometimes excessively so for my taste, and I would like to know what Glashow thinks about it. When people evoke Euclid’s axioms they usually refer to the one concerning the parallel lines. It was debated through the centuries, until it gave rise to non-Euclidian geometries. This proves that Euclid was right. This is indeed an independent axiom. Here I want to mention another axiom, the one which defines when two figures are equal. It says: “Και τα εφαρμόζοντα επ ́ άλληλα ίσα αλλήλοις εστίν,” or “The superposable objects (figures) are equal.” Euclid wanted to define when two geometrical figures, for example, two triangles, are equal. He postulates that they are equal if we can superpose one on the other. To proceed to this comparison we may have to perform two geometrical operations. The first is a translation, which means that we may have to move one of the triangles in order to bring its center to coincide with that of the other. The second is a rotation in order to check whether the two are exactly superposable. Obviously, we must assume that by applying either one of these two operations, the triangle does not change. So, Euclid implicitly assumes that this is an intrinsic property of the space, independent of the other axioms. If we want it, we must postulate it separately. Again, this notion of the isometries of space and time has a complicated history and I wonder whether Glashow intends to tell the story in one of the coming essays. I, for one, will be delighted to read it.

I will stop here. I am looking forward to reading the following essays and I anticipate the pleasure I will get. Thank you, Shelly.

John Iliopoulos

Sheldon Glashowreplies:

I very much enjoyed collaborating with John half a lifetime ago and I treasure our continuing friendship. Democritus was extraordinarily prescient to focus on atoms and empty space. Both notions were and remain seminal. Just this October we learned of a double neutron star merger whose behavior established the ultimate origin of gold and platinum atoms. The unsolved mystery of dark energy confirms Paul Davies’ remark that “the vacuum holds the key to a full understanding of the forces of nature.” Part II of my essay is done but gives short shrift to vacua other than Maxwell’s luminiferous aether. Fear not! The vacuum reappears when my narrative reaches quantum field theory and I must confront renormalization, spontaneous symmetry breaking, vacuum degeneracy and other noumena and phenomena. Plane geometry, especially the notion of axiomatic proof, made more sense to me than most other high-school courses. Only much later would I learn the elusive nature of truth in science. Euclid was, incidentally, first credited with recognizing the isometries of the plane by whichever nineteenth-century mathematician, perhaps Felix Klein, chose to refer to them as the Euclidean group.

John Iliopoulos is a CNRS Director of Research Emeritus at the Ecole Normale Supérieure in Paris, where for many years, he was the head of the Theoretical Physics Department.

Sheldon Lee Glashow is a Nobel Laureate and the Metcalf Professor of Mathematics and Physics at Boston University.